Geophys. J. Int. (2006) 166, 1224–1236 doi: 10.1111/j.1365-246X.2006.03030.x GJI Seismology Crosshole seismic waveform tomography – I. Strategy for real data application Yanghua Wang and Ying Rao Centre for Reservoir Geophysics, Department of Earth Science and Engineering, Imperial College London, South Kensington, London SW7 2BP, UK. E-mail: [email protected]Accepted 2006 April 1. Received 2006 April 1; in original form 2006 January 22 SUMMARY The frequency-domain version of waveform tomography enables the use of distinct frequency components to adequately reconstruct the subsurface velocity field, and thereby dramatically reduces the input data quantity required for the inversion process. It makes waveform tomog- raphy a computationally tractable problem for production uses, but its applicability to real seismic data particularly in the petroleum exploration and development scale needs to be ex- amined. As real data are often band limited with missing low frequencies, a good starting model is necessary for waveform tomography, to fill in the gap of low frequencies before the inversion of available frequencies. In the inversion stage, a group of frequencies should be used simultaneously at each iteration, to suppress the effect of data noise in the frequency domain. Meanwhile, a smoothness constraint on the model must be used in the inversion, to cope the effect of data noise, the effect of non-linearity of the problem, and the effect of strong sensitivities of short wavelength model variations. In this paper we use frequency-domain waveform tomography to provide quantitative velocity images of a crosshole target between boreholes 300 m apart. Due to the complexity of the local geology the velocity variations were extreme (between 3000 and 5500 m s −1 ), making the inversion problem highly non-linear. Nevertheless, the waveform tomography results correlate well with borehole logs, and provide realistic geological information that can be tracked between the boreholes with confidence. Key words: crosshole seismic, seismic inversion, waveform tomography. 1 INTRODUCTION Seismic waveform tomography especially when using transmission data is able to provide a quantitative image of physical properties in the subsurface, not only a structural image as in conventional seismic migration. It has the potential to image the velocity field with signif- icantly improved resolution, useful for time-lapse, high-resolution imaging of the reservoir. In crosshole seismic tomography, trav- eltime inversion uses first arrival times to reconstruct a velocity distribution of the survey region (Lytle & Dines 1980; McMechan 1983; Beydoun et al. 1989; Bregman et al. 1989; Washbourne et al. 2002; Bergman et al. 2004; Rao & Wang 2005). However, recorded seismic data contain not only first arrival time information but also scattered energy waveforms, not utilized in traveltime inversion. Waveform inversion attempts to use these waveforms for velocity model reconstruction (Pratt & Worthington 1990; Song et al. 1995; Pratt et al. 1998; Pratt 1999; Charara et al. 2000; Zhou & Greenhalgh 2003; Ravaut et al. 2004; Sirgue & Pratt 2004; Pratt et al. 2005). The process generally starts with an initial model and then updates it iteratively by minimizing the differences between the observed data wavefield and the theoretical data wavefield. This requests an efficient imaging tool, capable of being used on a production basis for practical problems. For waveform inversion, the frequency-domain version enables the use of distinct frequency components and thereby reduces the quantity of data required for processing (Pratt & Worthington 1990; Sirgue & Pratt 2004). Although in principle all frequencies may be modelled to fit the observations (equivalent to ‘time domain’ waveform inversion), in practice adequate reconstructions may be obtained with a reduced set of frequencies (‘frequency-domain’ waveform inversion). Marfurt (1984) pointed out that the frequency domain could be the method of choice for finite-difference/finite- element modelling if a significant number of source locations were involved. Pratt & Worthington (1990) pointed out that large aperture seismic surveys could be inverted effectively using only a limited number of frequency components. Recently Sirgue & Pratt (2004) further showed that frequency-domain inversion of reflection data using only a few frequencies could yield a result that is comparable to full time-domain inversion. To demonstrate this point, we set up a synthetic example, as shown in Fig. 1(a). The synthetic model we designed contains some real- istic geological features: channels, a fault, and a dipping layer. For crosshole traveltime tomography, it is almost impossible to recover a vertical structure with a sharp velocity change between the left and right (Bregman et al. 1989). We set up this extreme feature as an attempt to test the limits of waveform inversion. Traveltime 1224 C 2006 The Authors Journal compilation C 2006 RAS
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Geophys. J. Int. (2006) 166, 1224–1236 doi: 10.1111/j.1365-246X.2006.03030.xG
JISei
smol
ogy
Crosshole seismic waveform tomography – I. Strategy for real dataapplication
Yanghua Wang and Ying RaoCentre for Reservoir Geophysics, Department of Earth Science and Engineering, Imperial College London, South Kensington, London SW7 2BP, UK.E-mail: [email protected]
Accepted 2006 April 1. Received 2006 April 1; in original form 2006 January 22
S U M M A R YThe frequency-domain version of waveform tomography enables the use of distinct frequencycomponents to adequately reconstruct the subsurface velocity field, and thereby dramaticallyreduces the input data quantity required for the inversion process. It makes waveform tomog-raphy a computationally tractable problem for production uses, but its applicability to realseismic data particularly in the petroleum exploration and development scale needs to be ex-amined. As real data are often band limited with missing low frequencies, a good startingmodel is necessary for waveform tomography, to fill in the gap of low frequencies before theinversion of available frequencies. In the inversion stage, a group of frequencies should beused simultaneously at each iteration, to suppress the effect of data noise in the frequencydomain. Meanwhile, a smoothness constraint on the model must be used in the inversion, tocope the effect of data noise, the effect of non-linearity of the problem, and the effect of strongsensitivities of short wavelength model variations. In this paper we use frequency-domainwaveform tomography to provide quantitative velocity images of a crosshole target betweenboreholes 300 m apart. Due to the complexity of the local geology the velocity variations wereextreme (between 3000 and 5500 m s−1), making the inversion problem highly non-linear.Nevertheless, the waveform tomography results correlate well with borehole logs, and providerealistic geological information that can be tracked between the boreholes with confidence.
Figure 1. (a) A synthetic model consisting of vertical and dipping features for testing the waveform tomography approach. (b) Traveltime tomography result
which is used as the initial model for waveform tomography. (c) Reconstruction of the velocity image after using only the 200 Hz component in the tomographic
inversion. (d) The final reconstructed model after using eight selected frequencies between 200 and 900 Hz with 100 Hz interval.
Figure 2. (a) An example common-receiver gather at depth 2600 m that shows weak tube waves. (b) The common-receiver gather after f − k filtering for tube
wave attenuation. (c) An example common-shot gather at depth 2600 m which evidently shows strong tube waves. (d) The shot gather after f − k filtering for
tube wave attenuation.
transmission associated with the direct arrival. For comparison,
Fig. 4(b) shows modelled shot gather generated based on the fi-
nal tomography model. Although a frequency-domain waveform
inversion approach uses only a number of selected frequencies, the
data comparison reveals that the inversion model indeed is a good
representation of the subsurface earth model.
Fig. 5(a) is an example frequency slice (at 260 Hz) of the am-
plitudes of the real data set and, for comparison, Fig. 5(b) shows
Figure 3. (a) The amplitude spectrum of a typical shot gather at 2600 m (Fig. 2d). It has frequency bandwidth between 190 and 485 Hz. (b) Source wavelet
estimated from real data. It is used in waveform inversion.
Figure 4. (a) An example shot gather of the real data set after data windowing. (b) Modelled shot gather generated from the waveform inversion model. The
red curve is the first arrive time line picked from real data.
the corresponding frequency slice of the amplitudes of modelled
data set from waveform tomography. In the inversion, we start from
lower frequencies. For low frequencies the inversion method is more
tolerant of velocity errors, as these are less likely to lead to errors of
more than a half-cycle in the waveforms. As the inversion proceeds,
we move progressively to higher frequencies.
4 WAV E F O R M I N V E R S I O N
O F R E A L DATA
For the inversion of this real data set, we make the velocity model
discrete in cells with cell size 3 m to satisfy the criterion of four
cells per wavelength for the highest frequency (485 Hz) that we use
Figure 5. (a) A frequency slice of the amplitudes of the observed crosshole seismic data. (b) The frequency slice of modelled data, at the same frequency (260
Hz), generated from the final velocity model of waveform tomography.
in the inversion. The depth range that we choose to invert for is from
2497 to 3022 m. Therefore, there are altogether 101 rows and 176
columns in the grid.
To the beginning of waveform inversion scheme, an adequate
good starting model is necessary. This model should be capable of
describing the time domain data to within a half of the dominant
period, in order to avoid fitting the wrong cycle of the waveforms
(Pratt et al. 2005). The lower the frequency, the less accurate the
starting model needs be. However, all real data are band limited,
and thus a certain accuracy is required for the starting model. As
we have seen from Fig. 3, this real data set has a frequency gap
between 0 and 190 Hz. The lack of low-frequency information makes
the waveform inversion strongly depending on the initial model.
For waveform tomography we use the traveltime inversion result
as an initial model and proceed with waveform inversion using the
different frequencies.
Fig. 6(a) displays the initial model, the result of traveltime tomog-
raphy reported in Rao & Wang (2005). Fig. 6(b) is the ray density, the
(normalized) total length of ray segments across each single cell, a
direct indicator of the confidence in the traveltime inversion solution,
where a curved ray path is re-traced iteratively along with velocity
updating. This measurement of certainty, being proportional to the
ray density, can also be used in waveform tomography to build a
diagonal matrix C−1M , the inverse of model covariance matrix. The
latter is applied to the gradient vector γ before model updating (see
step 3 above).
In waveform tomography when dealing with real data, a model
smoothness constraint is a necessity. A number of real data exper-
iments we conducted indicate that, if we did not use smoothness
constraint in the inversion, waveform tomography would have not
converged at all, although we do not need such a constraint in syn-
thetic data examples (Fig. 1). Therefore, the primary cause is the
effect of data noise, which is not necessarily white in the frequency
domain. Strong outliers might have strong and biased influence on
the model update. As the frequency-domain data samples are com-
plex valued, it is not easy to mitigate the data noise in a way similar
to our method for winnowing traveltimes and amplitudes (Wang
et al. 2000). Waveform inversion is a highly non-linear problem but
if, in a linearized procedure, strong outliers are transferred linearly
to strong model updates, this causes the problem to be unstable and
divergent.
A second effect is the unevenly distributed ray density. As shown
in Fig. 5(b), the ray density distribution appears to be in short wave-
length variation. An uneven distribution of ray density will cause
a biased distribution of model update, as a model update is (in-
versely) proportional to the ray density through data residual back-
propagation. When constructing the model covariance matrix CM,
we could smooth the ray density distribution so as to change the
weight of model update. This approach might reduce the roughness
of model update for any single iteration and slow down the conver-
gence of the iterative procedure, but may not mitigate the problem in
the final solution. Ray density distribution is a measurement of the
illumination in the physical experiment, and thus reflects directly
the resolving power distribution.
The third effect is due to the model sensitivity. Investigation
in Wang & Pratt (1997) revealed that in traveltime inversion long
wavelength components of the velocity field are more sensitive than
the short wavelengths, and that in amplitude inversion short wave-
length components of the velocity field are much more sensitive
than the long wavelength components. Therefore, in waveform in-
version where amplitude information dominates, the data residual
tends to attribute to shortest wavelength components of the model
update first. This is contradictory to the philosophy of iterative lin-
ear inversion. In an iterative inversion, we must get the background
right first, so that linearization can be used for the inverse problem.
Some research groups have advocated amplitude-normalized wave-
form tomography, at least in the initial stages (Zhou & Greenhalgh
2003; Pratt et al. 2005).
This analysis suggests that we could use a smooth operator of
different size at each iteration, starting with a large smooth size and
then reducing the size gradually as iterations proceed. This approach
is sometimes referred to as a ‘multiscale’ approach. Pratt et al.
Figure 7. Waveform tomography experiment 1—the inversion is executed by each frequency consecutively. (a) The image after using five frequency components
between 190 and 210 Hz (with 5 Hz interval). (b) The result after using all 60 selected frequencies between 190 and 480 Hz. The image has strong X shaped
artefacts.
5 W E L L - L O G C O N S T R A I N E D
WAV E F O R M I N V E R S I O N
In this section, we use well-log information as a geological constraint
in the waveform inversion to test the dependence of the inversion
result on the initial model.
We design an initial model by combining sonic logging velocities
and the velocity field obtained from the traveltime tomography. The
velocity in the jth column of the initial model, v(init)j , is given by
v(init)j = w jv
(log)j + (1 − w j )v
(tt)j , (12)
where v(log)j is the logging velocity, v
(tt)j is the velocity obtained from
travel time tomography, and wj is the weighting coefficient. The
weight coefficient wj, as shown in Fig. 9, is set according to the
horizontal distance (Rao & Wang 2005).
We generate the well-log constrained initial model shown in
Fig. 10(a), where the logging velocity v(log)j for the initial model
building has been low-pass filtered. Then, using exactly the same
running parameters as those used in the experiment two above, we
obtain the well-log constrained velocity images, shown in Fig. 10(b).
In these images, the distinct layered structure with high/low veloc-
ities corresponds to the high- and low-velocity intervals in the well
logs.
Comparing the inversion result with and without well-log con-
straint (Figs 10b and 8b), we see that the inversion procedure that
we implemented has very weak dependency the well-logging con-
straint but depends strongly on the inversion strategy. The similarity
of two results in fact reveals the importance of the three issues we
discussed in the previous section for generating reliable inversion
results from real seismic data, and the importance of correctly set
up the initial model and the inversion strategy (using a group of
frequencies simultaneously and a model smoothness constraint).
In Fig. 11, we compare the traveltime inversion result (blue lines)
and waveform tomography result (red lines) both against velocity
curves from sonic logging in the boreholes (thin solid lines). We can
see clearly that the traveltime inversion result is the long wavelength
background velocity of the sonic logging and the waveform inver-
sion result. It indicates that the traveltime inversion result is indeed
a good initial model we use for waveform inversion.
6 C O N C L U S I O N S
In this paper we have discussed several practical issues in the ap-
plication of crosshole seismic waveform tomography when dealing
with real data. As real crosshole seismic data in most cases do not
contain low-frequency components (<190 Hz in the example pre-
sented), a good starting model is essential for the success of wave-
form tomography. At each inversion stage, it is also necessary to
invert a group of frequencies simultaneously to combat the effect
of data noise. We have demonstrated that the tomographic image
would be much better, in terms of interpretability, than that using
single frequency in sequence. In addition, a smoothness constraint
must be used in waveform tomography of real seismic data, to fur-
ther mitigate the effect of data noise and to combat the effect of
uneven distribution of ray density, the effect of the strong sensitivity
of short wavelength of velocity model, and the non-linearity of the
problem.
We have applied the frequency-domain waveform tomography
method to a real crosshole seismic data set acquired from two parallel
boreholes 300 m apart. After successful waveform tomography with
Figure 8. Waveform tomography experiment 2—inversion executed one group by one group in sequence. (a) The velocity image after using the first frequency
group (190, 195, . . . , 210 Hz). (b) The inversion result after using all 12 frequency groups; it is regarded as the final result of waveform tomography.
Figure 9. The diagrammatic curve of the relation between the weight coef-
ficient and the horizontal distance.
consideration of above three critical issues, we have also brought in
the sonic log information as a geological constraint in the inversion.
The result is similar to the inversion without the well-log constraint.
The reliability of waveform inversion without well-log constraint in
fact indicates the importance of the three issues in the waveform
inversion when we deal with the application of real seismic data.
A C K N O W L E D G M E N T S
Professors Gerhard Pratt, Albert Tarantola and Jeannot Trampert are
acknowledged for their constructive reviews on an earlier version
Figure 10. Logging constrained waveform tomography. (a) The initial model set as the mixture of well-log information of two boreholes and the traveltime
inversion result. (b) The final tomography result, after using all of the frequency groups in sequence.
Figure 11. Velocities from sonic logging in the boreholes (dark lines) compared with traveltime tomography (blue lines) and waveform tomography (red lines),
where (a) and (b) correspond to two boreholes, respectively.
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